Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Distribution of alternation points in uniform polynomial approximation


Author: G. G. Lorentz
Journal: Proc. Amer. Math. Soc. 92 (1984), 401-403
MSC: Primary 41A50; Secondary 41A10
DOI: https://doi.org/10.1090/S0002-9939-1984-0759662-2
MathSciNet review: 759662
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a continuous function $ f$ on $ \left[ {0,1} \right]$, we discuss the points where the polynomial $ {P_n}\left( x \right)$ of best uniform approximation deviates most from $ f\left( x \right)$, and the signs of the difference $ f\left( x \right) - {P_n}\left( x \right)$ alternate. We show that these points can be very irregularly distributed in $ \left[ {0,1} \right]$, even if $ f$ is entire.


References [Enhancements On Off] (What's this?)

  • [1] S. N. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, Gauthier-Villars, Paris, 1926.
  • [2] E. W. Cheney, Introduction to approximation theory, McGraw-Hill, New York, 1966. MR 0222517 (36:5568)
  • [3] H. Fiedler and W. B. Jurkat, Best $ {L^1}$-approximation by polynomials, J. Approximation Theory 37 (1983), 269-292. MR 693014 (84f:41006)
  • [4] M. I. Kadec, On the distribution of points of maximal deviation in the approximation of continuous functions by polynomials, Uspehi Mat. Nauk 15 (1960), 199-202. MR 0113079 (22:3920)
  • [5] J. H. B. Kemperman and G. G. Lorentz, Bounds for polynomials with applications, Indag. Math. 41 (1979), 13-26. MR 528214 (80f:30004)
  • [6] E. B. Saff and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie (L. Collatz et al., eds.), Internat. Schriftenreihe Numer. Math., no. 42, Birkhäuser Verlag, Basel, 1978, pp. 281-298. MR 527107 (80d:41008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A50, 41A10

Retrieve articles in all journals with MSC: 41A50, 41A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0759662-2
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society